"IID" or "iid" redirects here. For other uses, see: IID (disambiguation).

In probability theory, a sequence or other collection of random variables is independent and identically distributed (i.i.d.) if each has the same probability distribution as the others and all are mutually independent.

The abbreviation i.i.d. is particularly common in statistics (often as iid, sometimes written IID), where observations in a sample are often assumed to be (more-or-less) i.i.d. for the purposes of statistical inference. The assumption (or requirement) that observations be i.i.d. tends to simplify the underlying mathematics of many statistical methods. However, in practical applications this may or may not be realistic.

This is important in the classical form of the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution.

Examples

The following are examples or applications of independent and identically distributed (i.i.d.) random variables:

• All other things being equal, a sequence of outcomes of spins of a roulette wheel is i.i.d. From a practical point of view, an important implication of this is that if the roulette ball lands on 'red', for example, 20 times in a row, the next spin is no more or less likely to be 'black' than on any other spin.
• All other things being equal, a sequence of dice rolls is i.i.d.
• All other things being equal, a sequence of coin flips is i.i.d.
• One of the simplest statistical tests, the z-test, is used to test hypotheses about means of random variables. When using the z-test, one assumes (requires) that all observations are i.i.d. in order to satisfy the conditions of the central limit theorem.
• In signal processing and image processing the notion of transformation to IID implies two specifications, the "ID" part and the "I" part: (ID) the signal level must be balanced on the time axis, and (I=Independent) the signal spectrum must be flattened, i.e. transformed by filtering (such as deconvolution) to a white signal (one where all frequencies are equally present).

See also

• De Finetti's theorem
• Chebyshev's theorem